Binomial Coefficients And The Nature Of The Congruence

Binomial coefficients have been studied for 700 years, the sums of binomial coefficients have many good properties and are related to many mathematical problems. In 1978, the irrationality ofζ(2) andζ(3) was proved by Apéry using the recurrence for sums of the binomial coefficients. Since then, his work stimulated the interest of researching the sums of binomial coefficients, so a lot of new results are obtained. In this paper, several kinds of sums of binomial coefficients are mainly studied via the method of divisibility and congruence.The whole paper's content is as follows:Firstly the sums of binomial coefficients a_n(r,s) =(?) are considered in use of the method of congruence, the congruence properties of a_p(r,s), a_p-1(r,s), a_2p-1(r,s), a_p+1(r,s),a_kp(r, s) modulo p² are obtained.Secondly the congruence properties of u_a,b,c^ε{n) modulo p3 and modulo p², in what conditions p divides u_a,b,c^ε{n) and the connections of u_a,b,c^ε{n) withBernoulli numbers are discussed.Thirdly the congruence properties of (?) modulo p and some deductions are proved.These results will give people more effective information about these kinds of sum of binomial coefficients, which is useful for the further study of related problems.